The ExtendSeries(sol, deg) function returns the initial terms of the formal series solution sol extended to the specified formal degree deg.

The specified solution sol must be in the form returned by LinearFunctionalSystems[SeriesSolution] or ExtendSeries. In other words, sol must have an attribute of the special form as described in LinearFunctionalSystems[SeriesSolution].

This function computes the additional terms of the series expansions using the invertible leading matrix of the matrix recurrence system corresponding to the linear functional system that was originally specified. This recurrence is the part of the special structure stored in the attribute of the given solution.

The result of ExtendSeries is returned in the same form as the result of LinearFunctionalSystems[SeriesSolution] (the list of series expansions in x, corresponding to vars). The order term (for example, $\mathrm{O}\left(x^6\right)$) is the last term in the series. The result involves arbitrary constants of the form _c1, _c2, etc., and it has an attribute of the special form as described in LinearFunctionalSystems[SeriesSolution].

This function is part of the LinearFunctionalSystems package, and so it can be used in the form ExtendSeries(..) only after executing the command with(LinearFunctionalSystems). However, it can always be accessed through the long form of the command by using the form LinearFunctionalSystems[ExtendSeries](..).

$\mathrm{with}\left(\mathrm{LinearFunctionalSystems}\right)\:$

$\mathrm{sys}≔\left[\mathrm{diff}\left(\mathrm{y1}\left(x\right)\,x\right)-\mathrm{y2}\left(x\right)\,\mathrm{diff}\left(\mathrm{y2}\left(x\right)\,x\right)-\mathrm{y3}\left(x\right)-\mathrm{y4}\left(x\right)\,\mathrm{diff}\left(\mathrm{y3}\left(x\right)\,x\right)-\mathrm{y5}\left(x\right)\,\mathrm{diff}\left(\mathrm{y4}\left(x\right)\,x\right)-2\mathrm{y1}\left(x\right)-2x\mathrm{y2}\left(x\right)-\mathrm{y5}\left(x\right)\,\mathrm{diff}\left(\mathrm{y5}\left(x\right)\,x\right)-x^2\mathrm{y1}\left(x\right)-2x\mathrm{y3}\left(x\right)-\mathrm{y6}\left(x\right)\,\mathrm{diff}\left(\mathrm{y6}\left(x\right)\,x\right)-x^2\mathrm{y2}\left(x\right)+2\mathrm{y3}\left(x\right)\right]\:$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\,\mathrm{y3}\left(x\right)\,\mathrm{y4}\left(x\right)\,\mathrm{y5}\left(x\right)\,\mathrm{y6}\left(x\right)\right]\:$

$\mathrm{sol}≔\mathrm{SeriesSolution}\left(\mathrm{sys}\,\mathrm{vars}\right)$

$\mathrm{sol}≔\left[3{\mathrm{\_c}}_2+{\mathrm{\_c}}_5+\mathrm{O}\left(x\right)\,3{\mathrm{\_c}}_6+{\mathrm{\_c}}_3+\mathrm{O}\left(x\right)\,-\frac{{\mathrm{\_c}}_4}{2}+\mathrm{O}\left(x\right)\,{\mathrm{\_c}}_1+\frac{{\mathrm{\_c}}_4}{2}+\mathrm{O}\left(x\right)\,-{\mathrm{\_c}}_5+\mathrm{O}\left(x\right)\,{\mathrm{\_c}}_3+\mathrm{O}\left(x\right)\right]$

$\mathrm{ExtendSeries}\left(\mathrm{sol}\,5\right)$

$\left[3{\mathrm{\_c}}_2+{\mathrm{\_c}}_5+x\left(3{\mathrm{\_c}}_6+{\mathrm{\_c}}_3\right)+\frac{x^2{\mathrm{\_c}}_1}{2}+x^3{\mathrm{\_c}}_2+x^4\left(\frac{{\mathrm{\_c}}_3}{4}+\frac{{\mathrm{\_c}}_6}{2}\right)+\frac{x^5{\mathrm{\_c}}_1}{20}+\mathrm{O}\left(x^6\right)\,3x^2{\mathrm{\_c}}_2+x{\mathrm{\_c}}_1+{\mathrm{\_c}}_3+3{\mathrm{\_c}}_6+x^3\left({\mathrm{\_c}}_3+2{\mathrm{\_c}}_6\right)+\frac{x^4{\mathrm{\_c}}_1}{4}+\frac{x^5{\mathrm{\_c}}_2}{2}+\mathrm{O}\left(x^6\right)\,-\frac{{\mathrm{\_c}}_4}{2}-x{\mathrm{\_c}}_5+\frac{x^2{\mathrm{\_c}}_3}{2}+\frac{x^4{\mathrm{\_c}}_2}{4}+x^5\left(\frac{{\mathrm{\_c}}_6}{5}+\frac{{\mathrm{\_c}}_3}{10}\right)+\mathrm{O}\left(x^6\right)\,{\mathrm{\_c}}_1+\frac{{\mathrm{\_c}}_4}{2}+x\left(6{\mathrm{\_c}}_2+{\mathrm{\_c}}_5\right)+x^2\left(6{\mathrm{\_c}}_6+\frac{5{\mathrm{\_c}}_3}{2}\right)+x^3{\mathrm{\_c}}_1+\frac{9x^4{\mathrm{\_c}}_2}{4}+x^5\left(\frac{6{\mathrm{\_c}}_6}{5}+\frac{3{\mathrm{\_c}}_3}{5}\right)+\mathrm{O}\left(x^6\right)\,x^3{\mathrm{\_c}}_2+x{\mathrm{\_c}}_3-{\mathrm{\_c}}_5+x^4\left({\mathrm{\_c}}_6+\frac{{\mathrm{\_c}}_3}{2}\right)+\frac{3x^5{\mathrm{\_c}}_1}{20}+\mathrm{O}\left(x^6\right)\,x^3{\mathrm{\_c}}_6+x^2{\mathrm{\_c}}_5+x{\mathrm{\_c}}_4+{\mathrm{\_c}}_3+\frac{x^4{\mathrm{\_c}}_1}{4}+\frac{x^5{\mathrm{\_c}}_2}{2}+\mathrm{O}\left(x^6\right)\right]$

$\mathrm{sys}≔\left[\mathrm{y2}\left(x\right)x^2+3\mathrm{y2}\left(x\right)x+2\mathrm{y2}\left(x\right)-2\mathrm{y1}\left(x\right)x^2-4\mathrm{y1}\left(x\right)x+\mathrm{y1}\left(x+1\right)x^2+\mathrm{y1}\left(x+1\right)x\,\mathrm{y2}\left(x+1\right)-\mathrm{y1}\left(x\right)\right]\:$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\right]\:$

$\mathrm{sol}≔\mathrm{SeriesSolution}\left(\mathrm{sys}\,\mathrm{vars}\right)$

$\mathrm{sol}≔\left[{\mathrm{\_c}}_1+x\left(2{\mathrm{\_c}}_2+{\mathrm{\_c}}_1\right)+\mathrm{O}\left(x^2\right)\,x{\mathrm{\_c}}_1+\mathrm{O}\left(x^2\right)\right]$

$\mathrm{ExtendSeries}\left(\mathrm{sol}\,5\right)$

$\left[{\mathrm{\_c}}_1+x\left(2{\mathrm{\_c}}_2+{\mathrm{\_c}}_1\right)+x\left(x-1\right){\mathrm{\_c}}_2+\mathrm{O}\left(x^6\right)\,x{\mathrm{\_c}}_1+x\left(x-1\right){\mathrm{\_c}}_2+\mathrm{O}\left(x^6\right)\right]$